Download Chapter 12 Quantum Black Holes

Chapter 12
Quantum Black Holes
Classically, the fundamental structure of curved spacetime ensures that nothing can escape from within the
Schwarzschild event horizon. That is an emphatically deterministic statement. But what about quantum mechanics, which is fundamentally indeterminate?
12.1 Hawking Black Holes: Black Holes Are Not Really Black!
The uncertainty principle and quantum fluctuations of the
vacuum that play a central role in quantum mechanics. As
we now explain, because of quantum mechanics it is possible for a black hole to emit mass. Therefore, as Stephen
Hawking discovered, black holes are not really black!
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CHAPTER 12. QUANTUM BLACK HOLES
12.1.1 Deterministic Geodesics and Quantum Uncertainty
• Our discussion to this point has been classical in that it assumes
that free particles follow geodesics appropriate for the spacetime.
• But the uncertainty principle implies that
1. Microscopic particles cannot be completely localized on classical trajectories because they are subject to a spatial coordinate and 3-momentum uncertainty of the form ∆pi ∆xi ≥ h̄;
2. Neither can energy conservation be imposed except with an
uncertainty ∆E∆t ≥ h̄, where ∆E is an energy uncertainty
and ∆t is the corresponding time period during which this
energy uncertainty
• This implies an inherent quantum fuzziness in the 4-momenta
associated with our description of spacetime at the quantum level.
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For the Killing vector ξ ≡ ξt = (1, 0, 0, 0) in the
Schwarzschild metric,
2M
µ ν
0 0
,
ξ · ξ = gµν ξ ξ = g00ξ ξ = − 1 −
r
from which we conclude that
ξ is

 Timelike outside the horizon, since then ξ · ξ < 0,
 Spacelike inside the horizon, since then ξ · ξ > 0.
As we now make plausible, this property of
the Killing vector ξ permits a virtual quantum
fluctuation of the vacuum to be converted into
real particles that are detectable at infinity as
emission of mass from the black hole.
CHAPTER 12. QUANTUM BLACK HOLES
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Particle-antiparticle
fluctuation
Figure 12.1: Hawking radiation in Kruskal–Szekeres coordinates.
12.1.2 Hawking Radiation
Assume a particle–antiparticle pair created by vacuum fluctuation near
the horizon of a Schwarzschild black hole, such that the particle and
antiparticle end up on opposite sides of the horizon (Fig. 12.1).
• If the particle–antiparticle pair is created in a small enough region of spacetime, there is nothing special implied by this region
lying at the event horizon: the spacetime is indistinguishable
from Minkowski space because of the equivalence principle.
• Therefore, the normal principles of (special) relativistic quantum field theory will be applicable to the pair creation process
in a local inertial frame defined at the event horizon (even if the
gravitational field is enormous there).
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Particle-antiparticle
fluctuation
• In the Schwarzschild geometry, the conserved quantity analogous to the total energy in flat space is the scalar product of the
Killing vector ξ ≡ ξt = (1, 0, 0, 0) with the 4-momentum p.
• Therefore, if a particle–antiparticle pair is produced near the
horizon with 4-momenta p and p̄, respectively, the condition
ξ · p + ξ · p̄ = 0,
must be satisfied (to preserve vacuum quantum numbers).
• For the particle outside the horizon, −ξ · p > 0 (it is proportional
to an energy that is measureable externally).
• If the antiparticle were also outside the horizon, it too must have
−ξ · p̄ > 0, in which case
1. The condition ξ · p + ξ · p̄ = 0, cannot be satisfied
2. The particle–antiparticle pair can have only a fleeting existence of duration ∆t ∼ h̄/∆E (Heisenberg).
• So far, no surprises . . .
CHAPTER 12. QUANTUM BLACK HOLES
296
Particle-antiparticle
fluctuation
HOWEVER . . .
If instead (as in the figure) the antiparticle is inside the horizon,
• The Killing vector ξ is spacelike (ξ · ξ > 0)
• The scalar product −ξ · p̄ is not an energy (for any observer)
• In fact, the product −ξ · p̄ is a 3-momentum component.
• Therefore −ξ · p̄ can be positive or negative (!!).
Thus, there is magic afoot:
• If −ξ · p̄ is negative ξ · p + ξ · p̄ = 0 can be satisfied.
• Then the virtual particle created outside the horizon can propagate to infinity as a real, detectable particle, while the antiparticle remains trapped inside the event horizon.
• The black hole emits its mass as a steady stream of particles (!).
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Therefore, quantum effects imply that a black hole can
emit its mass as a flux of particles and antiparticles created
through vacuum fluctuations near its event horizon. The
emitted particles are termed Hawking radiation.
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CHAPTER 12. QUANTUM BLACK HOLES
The Hawking mechanism has been described by loose analogy with a
rather nefarious financial transaction
• Suppose that I am broke (a money vacuum), but I wish to give
to you a large sum of money (in Euros, E).
• Next door is a bank—Uncertainty Bank and Trust—that has lots
of money in its secure vault but has shoddy lending practices.
Then
1. I borrow a large sum of money ∆E from the Uncertainty
Bank, which will take a finite time ∆t to find that I have no
means of repayment.
We hypothesize ∆E · ∆t ∼ h, where h is constant, since the bank will be more diligent if
the amount of money is larger.
2. I transfer the money ∆E to your account.
3. I declare bankrupcy within the time ∆t, leaving the bank on
the hook for the loan.
A virtual fluctuation of the money vacuum has caused real money to
be emitted (and detected in your distant account!) from behind the
seemingly impregnable event horizon of the bank vault.
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12.1.3 Mass Emission Rates and Black Hole Temperature
The methods for obtaining quantitative results from the Hawking theory are beyond our scope, but the results can be stated simply:
• The rate of mass emission can be calculated using relativistic
quantum field theory:
h̄
dM
= −λ 2 ,
dt
M
where λ is a dimensionless constant.
• The distribution of energies emitted in the form of Hawking radiation is equivalent to a blackbody with temperature
h̄
h̄c3
M
⊙
T=
=
= 6.2 × 10−8
K.
8π kB M 8π kB GM
M
Advanced methods of quantum field theory are required to prove
this, but there are suggestive hints from the observations that
1. A black hole acts as a perfect absorber of radiation (as would
a blackbody).
2. Hawking radiation originates in random fluctuations, as we
would expect for a thermal emission process.
• Integrating dM/dt = −λ h̄/M 2 for a black hole assumed to emit
all of its mass by Hawking radiation in a time tH , we obtain
M(t) = (3λ h̄(tH − t))1/3 .
for its mass as a function of time.
CHAPTER 12. QUANTUM BLACK HOLES
Mass, Temperature
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Mass
Temperature
tH
Time
Figure 12.2: Evaporation of a Hawking black hole.
From the results
T=
h̄
8π kB M
h̄
dM
= −λ 2
dt
M
M(t) = (3λ h̄(tH − t))1/3
the mass and temperature of the black hole behave as in Fig. 12.2.
Therefore,
• The black hole evaporates at an accelerating rate as it loses mass.
• Both the temperature and emission rate of the black hole tend to
infinity near the end.
• Observationally we may expect a final burst of very high energy
(gamma-ray) radiation that would be characteristic of Hawking
evaporation for a black hole.
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12.1.4 Miniature Black Holes
How long does it take a black hole to evaporate by the quantum Hawking process? From the mass emission rate formula we may estimate
a lifetime for complete evaporation as
3
M3
−25 M
s.
tH ≃
≃ 10
3λ h̄
1g
• A one solar mass black hole would then take approximately 1053
times the present age of the Universe to evaporate (with a corresponding blackbody temperature of order 10−7 K). It’s pretty
black!
• However, black holes of initial mass ∼ 1014 g or less would have
evaporation lifetimes less than or equal to the present age of the
Universe, and their demise could be detectable through a characteristic burst of high-energy radiation.
• The Schwarzschild radius for a 1014 g black hole is approximately 1.5 × 10−14 cm, which is about 51 the size of a proton. To
form such a black hole one must compress 1014 grams (mass of
a large mountain) into a volume less than the size of a proton!
• Early in the big bang there would have been such densities.
Therefore, a population of miniature black holes could have formed
in the big bang and could be decaying in the present Universe
with a detectable signature.
• No experimental evidence has yet been found for such miniature black holes and their associated Hawking radiation. (They
would have to be relatively nearby to be seen easily.)
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CHAPTER 12. QUANTUM BLACK HOLES
12.1.5 Black Hole Thermodynamics
The preceding results suggest that the gravitational
physics of black holes and classical thermodynamics are
closely related. Remarkably, this has turned out to be correct.
• It had been noted prior to Hawking’s discovery that
there were similarities between black holes and black
body radiators.
• The difficulty with describing a black hole in thermodyanamical terms was that a classical black hole
permits no equilibrium with the surroundings (absorbs but cannot emit radiation).
• Hawking radiation supplies the necessary equilibrium that ultimately allows thermodynamics and a
temperature to be ascribed to a black hole.
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Hawking Area Theorem: If A is the surface area (area of
the event horizon) for a black hole, Hawking has proven
a theorem that A cannot decrease in any physical process
involving a black hole horizon,
dA
≥ 0.
dt
• For a Schwarzschild black hole, the horizon area is
A = 16π M 2
→
dA
= 32π M,
dM
which can be written (h = Planck, kB = Boltzmann).
kB A
h
d
dM =
8π kB M
4h
• But dE = dM is the change in total energy and the
temperature of the black hole is T = h̄/8π kB M.
• Therefore, we may write the preceding as
dE
| =
{zT dS}
1st Law
dS
≥ 0}
| {z
2nd Law
kB
S≡ A
| {z4h }
entropy
which are just the 1st and 2nd laws of thermodynamics if S ∝ (surface area of black hole) is entropy!
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CHAPTER 12. QUANTUM BLACK HOLES
Evaporation of a black hole through Hawking radiation
appears to violate the Hawking area theorem in that the
black hole eventually disappears. However,
• The area theorem assumes that a local observer always measures positive energy densities and that
there are no spacelike energy fluxes.
• As far as is known these are correct assumptions at
the classical level, but they may break down in quantum processes.
• The correct quantum interpretation of Hawking radiation and the area theorem is that
1. The entropy of the evaporating, isolated black
hole decreases with time (because it is proportional to the area of the horizon).
2. The total entropy of the Universe increases because of the entropy associated with the Hawking radiation itself.
3. That is, the area theorem is replaced by
Generalized 2nd Law: The total entropy of
the black hole plus exterior Universe may
never decrease in any process.
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12.1.6 Gravity and Quantum Mechanics: the Planck Scale
The preceding results for Hawking radiation are derived assuming
that the spacetime in which the quantum calculations are done is
a fixed background that is not influenced by the propagation of the
Hawking radiation.
• This approximation is expected to be valid as long as E << M,
where E is the average energy of the Hawking radiation and M
is the mass of the black hole.
• This approximation breaks down on a scale given by the Planck
mass
1/2
h̄c
MP =
= 1.2 × 1019 GeV c−2 = 2.2 × 10−8 kg.
G
• For a black hole of this mass, the effects of gravity become important even on a quantum (h̄) scale, requiring a theory of quantum gravity.
• We don’t yet have an adequate theory of quantum gravity.
• Note that near the endpoint of Hawking black hole evaporation
one approaches the Planck scale.
• Therefore, we do not actually know yet what happens at the conclusion of Hawking evaporation for a black hole.